= Solution
The <character of a representation> is the <trace> of its representing matrix. For the rotation powers,
$$
\chi(U^j)=1+2\cos(j\pi/2).
$$
Each $U^jV$ has a planar reflection block of trace zero and third diagonal entry $-1$, giving character $-1$. In the chosen element order,
$$
\boxed{(\chi(g_0),\ldots,\chi(g_7))=(3,1,-1,1,-1,-1,-1,-1).}
$$
Equivalently its values on the <conjugacy classes> $\{I\}$, $\{U^2\}$, $\{U,U^3\}$, $\{V,U^2V\}$, $\{UV,U^3V\}$ are $3,-1,1,-1,-1$.
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